%C T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.

%C The limit of the diagonals is A000712 (partitions into parts of two kinds). In particular, if 0<=m<=n, T(n(n+1)/2 + m, n) = A000712(m). These partitions in this range can be viewed as an equilateral right triangle of side n, with one partition appended on the top (at the left) and another appended on the right. - _Franklin T. Adams-Watters_, Jan 11 2006

%F T(n,k) = Sum_{i=0}^{n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.

%e T(5,2) = 2 because the only partitions of 5 having Durfee square of size 2 are [3,2] and [2,2,1]; the other five partitions ([5], [4,1], [3,1,1], [2,1,1,1] and [1,1,1,1,1]) have Durfee square of size 1.

%e Triangle starts:

%e 1;

%e 2;

%e 3;

%e 4, 1;

%e 5, 2;

%e 6, 5;

%e 7, 8;

%e 8, 14;

%e 9, 20, 1;

%e ...

%p g:=sum(t^k*q^(k^2)/product((1-q^j)^2,j=1..k),k=1..40): gser:=series(g,q=0,32): for n from 1 to 27 do P[n]:=coeff(gser,q^n) od: for n from 1 to 27 do seq(coeff(P[n],t^j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form